Quartile

One of the four divisions of observations which have been grouped into four equal-sized sets based on their statistical rank. The quartile including the top statistically ranked members is called the first quartile and denoted . The other quartiles are similarly denoted , , and . For  data points with  of the form  (for , 1, ...), the hinges are identical to the first and third quartiles.

The following table summarizes a number of common methods for computing the position of the first and third quartiles from a sample size  (P. Stikker, pers. comm., Jan. 24, 2005). In the table,  denotes the nearest integer function.

method	1st quartile	1st quartile	3rd quartile	3rd quartile
	odd	even	odd	even
Minitab
Tukey (Hoaglin et al. 1983)
Moore and McCabe (2002)
Mendenhall and Sincich (1995)
Freund and Perles (1987)

REFERENCES:

Freund, J. and Perles, B. "A New Look at Quartiles of Ungrouped Data." American Stat. 41, 200-203, 1987.

Hoaglin, D.; Mosteller, F.; and Tukey, J. (Ed.). Understanding Robust and Exploratory Data Analysis. New York: Wiley, pp. 39, 54, 62, 223, 1983.

Kenney, J. F. and Keeping, E. S. "Quartiles." §3.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 35-37, 1962.

Mendenhall, W. and Sincich, T. L. Statistics for Engineering and the Sciences, 4th ed. Prentice-Hall, 1995.

Moore, D. S. and McCabe, G. P. Introduction to the Practice of Statistics, 4th ed. New York: W. H. Freeman, 2002.

Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 184-186, 1967.